Probing the Variation of the Intervalley Tunnel Coupling in a Silicon Triple Quantum Dot

Electrons confined in silicon quantum dots exhibit orbital, spin, and valley degrees of freedom. The valley degree of freedom originates from the bulk band structure of silicon, which has six degenerate electronic minima. The degeneracy can be lifted in silicon quantum wells due to strain and electronic confinement, but the “valley splitting” of the two lowest-lying valleys is known to be sensitive to atomic scale disorder. Large valley splittings are desirable to have a well-defined spin qubit. In addition, an understanding of the intervalley tunnel coupling that couples different valleys in adjacent quantum dots is extremely important, as the resulting gaps in the energy-level diagram may affect the fidelity of charge- and spin-transfer protocols in silicon quantum-dot arrays. Here we use microwave spectroscopy to probe variations in the valley splitting, and the intra- and intervalley tunnel couplings ($t_{ij}$ and $t_{ij}^{\prime }$) that couple dots $i$ and $j$ in a triple quantum dot. We uncover large variations in the ratio of intervalley to intravalley tunnel couplings $t_{12}^{\prime }/t_{12}=0.90$ and $t_{23}^{\prime }/t_{23}=0.56$. By tuning the interdot tunnel barrier we also show that $t_{ij}^{\prime }$ scales linearly with $t_{ij}$, as expected from theory. The results indicate strong interactions between different valley states on neighboring dots, which we attribute to local inhomogeneities in the silicon quantum well.

Popular Summary

Impressive progress has been made in the ability to coherently control the spin states of single electrons confined in silicon quantum dots. However, the presence of valleys, an additional orbital degree of freedom of electrons in silicon, may present significant challenges when scaling up the number of silicon qubits. Recently, the hybridization of semiconductor quantum dots and superconducting cavities has allowed coherent coupling between single-electron spins and microwave photons. Here, we make use of this hybrid-device architecture to sensitively probe the spatial variation of valley states in a silicon triple quantum dot. Our characterization approach may help to improve $\mathrm{Si}$/${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ heterostructure growth and yield more reliable silicon quantum devices.

The spin state of a single electron can be harnessed to yield a robust two-level quantum system for quantum information processing. However, in silicon, valley states result in additional unwanted degrees of freedom that are generally difficult to control and affect the quality of spin operations. Most characterization methods focus on identifying valley-state energies. However, the unexplored intervalley tunnel coupling between neighboring quantum dots is important for scaled applications where single electrons will be shuttled between dots in a spin-qubit array. By measuring the transmission through a microwave resonator coupled to an array of three quantum dots, we probe not only the valley-state energies in the array, but also their interactions with each other.

Our results indicate that there are highly localized variations in the substrate quality of quantum-dot devices, leading to different valley-state behavior for each quantum dot. We hope that our valley-state characterization method, in combination with continued materials development, will yield more uniform valley properties in future silicon spin-qubit devices.

Figure 1

Valley physics in silicon. (a) First Brillouin zone of silicon, with six degenerate conduction-band minima. Tensile strain in the $\mathrm{Si}$ quantum well separates the $\pm z$ valleys from the $\pm x,y$ valleys, while vertical confinement lifts the degeneracy of the $\pm z$ valleys. (b) TEM image of the $\mathrm{Si}$/${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ quantum well. (c) Schematic cross section of a DQD with corresponding energy levels. An ideal $\mathrm{Si}$/${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ quantum well with abrupt interfaces (left) leads to large and uniform valley splittings, and no intervalley tunnel coupling, while a realistic quantum well with soft interfaces and step edges (right) will have small, nonuniform valley splittings, and strong intervalley tunnel coupling between the quantum dots. Corresponding energy-level diagrams are shown in the bottom graphs.

Figure 2

Cavity-coupled TQD. (a) Optical image of a superconducting cavity coupled to two TQDs. (b) Schematic of the TQD device with cavity coupler gate (CP). (c) False-color SEM image of the TQD. (d) Large-scale charge stability diagram of the TQD. The dashed black lines indicate the three quantum-dot charge transitions. Features corresponding to two-level systems (TLSs) are indicated by black arrows. (e) Charge stability diagram in the single-electron regime. The blue star indicates the (1,0,0)-(0,1,0) interdot transition and the red triangle indicates the (0,1,0)-(0,0,1) interdot transition. (f) Schematic of the operating points indicated by the symbols in (e). Due to the geometry of CP, the microwave coupling to dot 3 is strongest, as indicated by the thickness of the black arrows.

Figure 3

Cavity response in the single-electron regime at $T_e=350\mathrm{mK}$. (a) Energy bands of the TQD, with valley splittings denoted by $E_{\mathrm{VS},i}$. At low temperatures, only the transition between the lowest energy states (blue) is visible in the cavity response. By increasing temperature, intervalley transitions (red) can be accessed. (b) Cavity response $A/A_0$ near the (1,0,0)-(0,1,0) transition as a function of $t_{12}$ and the detuning ${ϵ}_{12}$ between dots 1 and 2. The colored arrows correspond to the transitions highlighted in (a). (c) $A/A_0$ in the vicinity of the (0,1,0)-(0,0,1) transition as a function of $t_{23}$ and ${ϵ}_{23}$. Insets in (b),(c) show the simulated cavity response.

Figure 4

Quantitative analysis of the cavity response. (a) Energy-level diagram for $|t_{12}|/h=2.9\mathrm{GHz}$. Transitions $\mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\mathrm{\prime }}$ are energetically accessible at $T_e=350\mathrm{mK}$. (b) Transition frequencies $\mathrm{\Omega }/h$ and ${\mathrm{\Omega }}^{\mathrm{\prime }}/h$ as a function of ${ϵ}_{12}$ for the four values of $t_{12}$ indicated in (c). The minima in the transition frequencies are labeled (i)–(iii). (c) Line cuts extracted from Fig. 3 for different $|t|$ (fits are shown in black). The line cuts correspond to the low $|t|$ regime, resonant regimes for both intra- and intervalley transitions, and the high $|t|$ regime. (d) Extracted intervalley tunnel coupling rates $|t_{ij}|/h$ as a function of $|t_{ij}|/h$ for both interdot transitions.

Figure 5

Single-electron occupation of the TQD. (a) Cavity transmission $A/A_0$ as a function of plunger gate voltages $V_{\mathrm{P1}}$ and $V_{\mathrm{P2}}$, showing charge stability diagram of dots 1 and 2 with dot 3 empty. (b) $A/A_0$ as a function of $V_{\mathrm{P2}}$ and $V_{\mathrm{P3}}$, showing charge stability diagram of dots 2 and 3 with dot 1 accumulated to extend the reservoir.

Figure 6

Tunnel coupling calibration in the TQD. (a) $t_{12}$ is extracted as a function of $V_{\mathrm{B2}}$ by fitting the input-output model described in the main text to line cuts of Fig. 3 data. (b) Similarly, $t_{23}$ is extracted as a function of $V_{\mathrm{B3}}$ using line cuts of Fig. 3 data.